Silence is one of the most under-used teaching techniques. As Julia learns mathematics, I notice major differences in the way my husband and I respond to her. After I ask her a question, I wait for an answer. The period at the end of that sentence is deliberate. I don’t do anything else. I don’t give her any prompts or hints. If she whines that she can’t get it, I tell her to keep thinking about it. If she comes up with the wrong answer, I tell her that it’s wrong and she should try again. Almost always, she can find the mistake she made.
Dennis, like most people, will try to help her if she doesn’t answer right away, by giving her a hint. Often that makes it more difficult to solve the problem because she now has the original problem to solve plus trying to figure out how the hint relates to it, not an easy task for a fourth-grader. Alternatively, he will give her the answer and then tell her to try the next problem, which is always just like the previous problem, that being the way math textbooks in America are structured. Since she could not figure out the previous problem, she is not going to get this one, either.
Dennis has degrees in Mathematics and Physics from UCLA. He was an excellent student in math and he acts the way his teachers acted in school. Paradoxically, this is not the way he learned mathematics. He had taught himself Calculus by the eighth grade from books he checked out of the public library.
My three recommendations for anyone who wants to be a better math teacher.
- Give students fewer problems.
- Give them the time to solve those problems on their own.
- Be quiet and let them do it.
Sites I liked today on teaching Algebra
Purple Math - I especially liked their “how do I really do this stuff” lessons. Readable and easy to understand. Also recommended for adults who knew they once knew, e.g. what a negative exponent was. Those of you who have not had a math class in years can peruse this site for lots of those moments when you smack your forehead and say, “Oh,yeah, THAT’S what that is.”
Teaching College Math Technology Blog – offers thoughts on demonstrations, learning activities and the use of technology.
The Wolfram Demonstrations Project is way cool - I say this being full aware of the fact that if there is such a thing as a visual learner, I am not it. You can download the Mathematica player for free and run anyone of their demonstrations. Be aware that even with high-speed access the player takes a long time to download. Be patient.
When I look at the wealth of resources, from the straightforward, readable pages on Purple Math to the high-tech demonstrations of the Wolfram Project, it is hard to believe that every math class in this country is not an amazing place to learn. One reason why is that after teachers have finished teaching, tutoring students after school, grading papers and preparing for the next day’s lesson, they just don’t have time. In the summer, far too many are painting houses, teaching summer school or other second jobs just to make ends meet.
I really do think one solution for teachers, just like for Julia, is providing time and silence. If we paid our teachers for those two months in the summer to come in and work on making their mathematics classes better, I wonder how our schools would change for the better.